A Multiple Comparison Procedure for Populations with Unequal Variances

Transcription

1 Journal of Statistical Theory and Applications Volume 11, Number 2, 212, pp ISSN A Multiple Comparison Procedure for Populations with Unequal Variances Hong Li Department of Mathematical Sciences, Cameron University, Lawton, OK 7355, USA Abstract This article investigates an exact method that extends Dunnett s method on the multiple comparisons with a control (MCC) to the case of unequal error variances when the ratios of population variances of k new treatments to that of the control group are known from previous experience. The advantages of the proposed method is illustrated through an example. The simulation results show that the typical methods assuming the equal variance will have inflated error rate and may lead to erroneous inference. In addition, two approximate approaches are proposed to provide the simultaneous confidence intervals for the differences between each new treatment and the control mean with unknown ratios and unequal variances. The Monte Carlo study shows that the proposed approach always controls the family-wise error rate at a specified nominal level α, while some established method has inflated error rate in some instances. Keywords and phrases: Multiple comparisons with a control; Family-wise error rate; Simultaneous confidence intervals; Unequal variances. AMS 2 subject classification: 62F99 Corresponding author. lhong@cameron.edu

2 H. Li Introduction For the problem of comparing means of two normal populations with unequal variances, a large number of approximate tests and exact tests are available in the literature. Fisher (1935) proposed the Fiducial approach for obtaining a probability distribution of a parameter from the observed data. Tsui and Weerahandi (1989) considered a generalized test to the Behrens-Fisher problem. Welch [15, 16] proposed an approximate t test when the variances of two samples are different. However, there has been only a few attempts to extend these results to the problem of testing the equality of a number of means when the population variances are not homogeneous [11, 12, 13]. Tamhane [12] proposed two approximated approaches for the multiple comparisons with a control and all-pariwise comparisons when the variances are unequal. Games and Howell (1976) provided the approximate simultaneous confidence intervals for all-pairwise differences under the heteroscedasticity. Hochberg (1976) proposed an approximate approach for all-pairwise comparisons. In a phase II clinical study, multiple groups with different dose levels are usually of interest to be compared with a control group to detect the effective size of the dose level [14]. In such a circumstance, the comparison of the primary interest is the comparison of new treatments with a control. The parameters of interest are µ i µ for i = 1,..., k, the difference between each new treatment mean µ i and the control mean µ. Usually, the condition of equal variance is assumed across all treatment and control groups, which legitimized the one-way ANOVA assumption. One of the main goals of phase II clinical study is to estimate the minimum effective dose (MED), which is the lowest dose level for which the response exceeds that of the control group by a prespecified threshold amount and the maximum safe dose (MSD). In [1, 3, 5, 1], some step-wise multiple test procedures have been proposed to identify the MED and/or MSD, while they do not provide the simultaneous confidence intervals for the differences in mean. Under the usual normality and equality of variance assumption, Dunnett [2] provided the simultaneous confidence lower bounds, upper bounds and confidence intervals for µ i µ, i = 1,..., k, which have been widely used to estimate the MED and/or MSD in dose-response studies. In practice, however, the homogeneity of variances is seldom satisfied. For instance, in a phase II clinical trial [9], a new drug for treatment of arthritis of the knee with four dose levels and a placebo control group are compared to select the MED and MSD. The efficacy variable is the pooled

3 A Multiple Comparison Procedure 167 WOMAC (Western Ontario and McMaster University Osteoarthritis Index) score, which is a composite score computed from the assessments of pain, stiffness and physical function. An increase in WOMAC indicates amprovement in disease condition. The safety variable is the serum level of a certain chemical. If the variances across all five treatment groups are different for both efficacy and safety measurements, Dunnett s method may provide larger confidence lower bounds on mean differences for efficacy. This may lead to ancorrect decision. An ineffective dose may be claimed as effective and be selected into the phase III trial. Meanwhile, Dunnett s method may provide smaller confidence upper bounds for the safety, which will lead to erroneous inference. An unsafe dose may be stated as safe and proceed to the next stage, which will be harmful for the patients who participate in the trial. Therefore, failure of the assumption of equal variances can be a serious problem in biomedical experiments, and one can not afford to ignore the problem for the simplicity of methods that assume the equality of variances. In this paper, we propose an exact and two approximate methods for MCC with and without the knowledge of ratios of population variances when the equality of variances can not be assumed. This article is organized as follows. The exact method is proposed in Section 2. An applicatios given Section 3 to illustrate the advantage of the proposed method compared to Dunnett s method under the assumption of the heterogeneity of the variances. A simulation study is carried out to compare the error rate of the proposed method and Dunnett s method, which is given Section 4. In Section 5, we propose two approximate approaches to estimate the critical values that provide the simultaneous confidence intervals for MCC when the ratios of variances are unknown. We conduct a Monte Carlo study to compare the error rate between the proposed approach and Tamhane s (1977) approach in Section 6. The discussion of the results is given Section 7. 2 An Exact Method when Ratios of Variances are Known From a practical point of view, the variation of responses under different dose levels is usually different with the change of dose levels because patients in different groups tend to respond differently due to some biological factors or the toxicity effect at various dose levels. We are motivated to consider a simultaneous confidence interval method when the equal variance as-

4 H. Li 168 sumptios not satisfied. In this section, we assume that the ratios of the population variances between each dose group and that of the control group are known from the prior experience. Suppose the Y ij is the observed measurement of the jth subject in the ith treatment, and the Y ij s are independently distributed as N(µ i, σi 2 ). Consider the unbalanced one-way model, Y ij = µ i + ε ij (1) with i =, 1,..., k, j = 1, 2,...,, where ε ij s follow N(, σ2 i ), i = refers to the control group (either placebo or active control), and i = 1, 2,..., k refer to k different dose groups. Denote the sample mean measurement by Ȳi, which is the least square estimate for µ i. Denote the sample variance by Si 2, which is given by S 2 i = 1 1 j=1 (Y ij Ȳi) 2, where S 2 i σ2 i v i χ 2 v i, (v i = 1), and is independent of Ȳ i. Denote the ratio of population variance of responses from the ith dose group to that of the control group by λ i, i.e., λ i = σ 2 i /σ2 for i =, 1,..., k, with λ = 1. Let σ 2 be the variance of the response of the control group, and σ 2 i (i = 1,..., k) stands for the variance of the response of the ith dose group. Consider testing the differences between two population means, H i : µ i µ δ vs. H ai : µ i µ > δ, with i = 1,..., k, where δ is a given threshold constant. The test statistic is constructed as t i = ȳi ȳ δ ˆσ λi + 1, i = 1,..., k, (2) where t i has a student t distribution with v degrees of freedom (v = k i= (k + 1)). ˆσ is a pooled estimator for the standard deviation of the control group and given by ˆσ = 1 v k i= S 2 i ( 1) λ i. It is straightforward to show that ˆσ is an unbiased estimate of σ. Let d be the common critical constant of the multivariate t distribution and it is the solution to the equation P ȳi ȳ (µ i µ ) ˆσ λi + 1 < d, i = 1,..., k = 1 α. (3)

5 A Multiple Comparison Procedure 169 The integration form for equation (3) can be expressed as k where r = ˆσ σ, p(r) is the density function of a [ ni Φ z + d (r) 1 + n ] i φ(z)p (r) dzdr = 1 α (4) λ i λ i χ 2 v v random variable. Φ is the standard normal distribution function, φ(z) is the probability density function of a standard normal random variable. Therefore, the 1(1 α)% simultaneous confidence lower bounds for µ i µ are given by µ i µ > ȳ i ȳ dˆσ λi + 1 for i = 1,..., k. (5) Similarly, the 1(1 α)% simultaneous confidence upper bounds for µ i µ are given by µ i µ < ȳ i ȳ + dˆσ λi + 1 for i = 1,..., k. (6) For two-sided multiple comparisons with a control, the simultaneous confidence interval for the difference between each new treatment mean µ i and the control mean µ is given by where d is the solution to the equation k µ i µ ȳ i ȳ ± d ˆσ λi + 1 for i = 1,..., k, (7) [ ni Φ(z + d (r) 1 + n i ni ) Φ(z d (r) 1 + n ] i ) λ i λ i λ i λ i φ(z)p(r)dzdr = 1 α (8) 3 An Example The proposed method can be applied to does-response studies to detect the effective and safe doses when the variances are different among different dose groups and ratios of variances are available. In this section, we illustrate the advantage of the proposed method by applying it to a real data, which is about the human food safety of recombinant insulin-like growth factor-i (rigf I). The data was used in [6] to illustrate a multiple comparison procedure. It comes from an experiment in which absolute weights of organs are measured from control hypophysectomized rats and hypophysectomized rats treated orally with hormone rigf-i. One group is given saline control; another group is given bovine serum albumin (BSA) as a negative

6 H. Li 17 oral protein control; four other groups were given rigf-i with the different dosages. The rats spleens are treated for either 17 days by gavage or 15 days by continuous subcutaneous (SC) infusion. The data is summarized in Table 1 (see appendix). We will concentrate on comparing five treatment groups with the saline control group. Suppose that we are interested in finding if the treatment increases the liver weight of rats. Without loss of generality, assume δ =, the test can be set up as H i : µ i µ vs. H ai : µ i µ >, with i = 1,..., 5, where µ is the mean weight of the saline control group, µ i refers to the mean weight of the ith treatment group. The Levenue test for the equality of variances yields a significant result (F=12.28, p-value<.1), which indicates that the equal variance assumptios invalid. From the prior experience, the ratios of variances are known and are given by λ 1 =... = λ 4 =.8, λ 5 = 1.5. The 95% confidence lower bounds for the exact method and Dunnett s method are given Table 2 (see appendix). The result indicates that dose 5 significantly increases the mean spleen weight. However, the lower bound of mean difference between the SC infusion group (dose 5) and saline control group is under equal variance assumption, which is larger than the one obtained from the exact method (55.3). It should be noticed that when the equal variance assumptios not valid, Dunnett s method may lead to an erroneous inference on the difference of mean weight between the SC infusion and saline control groups. As a consequence, aneffective dose may be incorrectly stated as effective. For example, if the threshold constant is 6 instead of, that is, we are testing H i : µ i µ 6 vs. H ai : µ i µ > 6, with i = 1,..., 5. Based on the exact method, none of the hypotheses will be rejected, it concludes that none of the five treatments will significantly increase the spleen weight by 6 grams. On the other hand, H 5 is rejected based on Dunnett s method. It will falsely conclude that the SC infusion significantly increases the spleen weight by 6 grams. 4 A Simulation Study on Error Rate We conduct a simulation study to show that Dunnett s method may have excessive error rates when the equal variance assumptios not satisfied. We generate three random samples from a normal distribution with mean µ = 3 and different variances. We compute the error rate [4], the probability of rejecting at least one hypothesis given all the hypotheses are true when the

7 A Multiple Comparison Procedure 171 ratios λ i are.5, 1, 1.5, 2, 2.5 and 3 respectively. For the convenience, we assume the ratios are the same for all the treatment groups. The estimated error rates are given Table 3 (see appendix) based on 5 simulations. The standard errors of the estimates are given the parentheses (the second row). We also compute the 95% confidence intervals for the true error rate (ˆp ± 1.96 ˆp(1 ˆp)/5) for each λ i, which are given the third and fourth row in Table 3. When the variances are same (λ i = 1), the estimated error rates are the same for both methods. It indicates that both methods can control the error rate when the variances are equal. When the variances are different among the control and new treatments, for Dunnett s method, all the confidence intervals for the error rate do not contain the nominal level.5. The confidence intervals for the error rate for the exact method, however, all contai.5 for different values of λ. This indicates that Dunnett s method can not control the error rate when the equal variance assumption fails. 5 Multiple Comparisons with a Control when the Ratios of Variances are Unknown When the prior knowledge of the ratios of variances is unavailable, we propose two approximate procedures to estimate the critical values and provide the confidence intervals for µ i µ. Consider the unbalanced one-way model given Section 2. Let i = be the index for control group, and i = 1, 2,..., k be the indices for each new treatment. Suppose we are interested in comparing each new treatment mean with the control mean. Let λ i = σi 2/σ2, i = 1,...k, denote the ratios of variances of the ith new treatment to that of the control group, which are unknown. The ratios can be estimated by ˆλ i = cs 2 i /S2, i = 1,..., k, where c = ( 3)/( 1). ˆλ i is an unbiased estimator of λ i according to Lehmann [7]. Consider testing the hypotheses, H i : µ i µ δ vs. H ai : µ i µ > δ, with i = 1,..., k, and δ is a given threshold constant. The test statistic is considered as t i = ȳi ȳ δ ˆσ ˆλi c + 1 (9) = ȳi ȳ δ, S 2 i + S2 i = 1,..., k, (1)

8 H. Li 172 where ˆσ = S, S 2 and S2 i are the sample estimates of variances of the control and the ith new treatment group. The 1(1 α)% simultaneous confidence lower bounds for µ i µ are given by µ i µ > ȳ i ȳ q S 2 i + S2 for i = 1,..., k, where the critical value q is the solution to the following equation ȳ P (µ i µ ) ˆσ ˆλi c + 1 < q, i = 1,..., k = 1 α, (11) in order to guarantee the overall coverage probability to be 1 α. Based on Welch s (1938) method and the test statistic given (1), Tamhane s (1977) method provides the approximate 1(1 α)% simultaneous confidence intervals for µ i µ, which is given by µ i µ ȳ i ȳ tˆvi,β(s 2 i + S2 ) 1/2, i = 1,..., k, where β = 1 (1 α) 1/(k 1), ˆv i is given by ˆv i = ( ˆσ 2 i + ˆσ2 ) 2 ˆσ 4 i n 2 i ( 1) + ˆσ 4 n 2 ( 1) (12) Tamhane s method has a joint confidence level less than 1 α in some cases, which will result in inflated family-wise error rate. We estimate the error rate by Monte Carlo studies in the next section. Here, we will estimate the critical value based on the test statistic given (9). The probability given (11) can be written as the following integration form, G(z 1, z 2,, z k )p(γ )f(a 1 ) f(a k )d(γ )da 1 da k, where z i = q ˆσ σ ai, G(z 1,..., z k ) is the cdf of the multivariate normal distribution. γ = ˆσ σ, p(γ ) is the density function of a χ 2 v v variable random (v = 1), and a i = ( ˆλ i c + 1 )/( λ i + 1 ). It can be shown that a i approximately follows F distribution (see appendix) with ˆf i and v degrees of freedom based on Welch s (1938) method, and ˆf i can obtained by (12). f(a i ) is the density function of a i, it is given by f(a i ) = 1 a i B( ˆf i 2, v 2 ) ( ˆf i a i ) ˆf iv v ( ˆf i a i + v ) ˆf i +v.

9 A Multiple Comparison Procedure 173 Therefore, q is the solution to G(z 1, z 2, z k )p(γ )f(a 1 ) f(a k )d(γ )da 1 da k = 1 α. (13) Since the correlation between z i and z j depends on the unknown σi 2 and σ2, and it is difficult to implement the multiple integration when k is large, we propose two approximate procedures to estimate the critical value. The first one is using Bonferroni s approximation. Instead of a common critical value q, we will find a critical value q i for each comparison. In specific, q i can be estimated by solving the following equation Φ (q i (γ ) a i ) p(γ )f(a i )d(γ )da i = 1 α k, where Φ is the cdf of the standard normal distribution. A set of conservative confidence lower bounds on µ i µ are given by µ i µ > ȳ i ȳ q i S 2 i + S2 for i = 1,..., k. The second procedure is based on Slepian s inequality, which is given the following Theorem. Theorem 5.1 (Slepian, 1962) Let Z 1,..., Z p be multivariate normal random variables such that for all i j; then for any constants c 1,..., c p, Cov(Z i, Z j ) P {Z i c i for i = 1,..., p} p P {Z i c i }. Note that cov(z i, z j ) > for all i j. By applying Theorem 5.1 (see appendix), an approximate solution for the critical value q can be obtained by solving the equation k where Φ is the cdf of the standard normal distribution. Φ(q(γ ) a i )p(γ )f(a i )d(γ )da i = 1 α, (14) 6 A Monte Carlo Study on Error Rate We carry out a simulation study to compare the error rate of three methods for multiple comparisons with a control. One of those is based on Slepian s inequality proposed in the previous

10 H. Li 174 section. The second one is Tamhane s (1977) method based on Welch s approximation, and we call it TM1. The third one is Tamhane s (1977) method based on Banerjee s solution to Behrens-Fisher problem, and we call it TM2. We generate three random samples from a normal distribution with mean µ = 1 and different variances. The error rate is defined the same as in Section 4. We assume that the ratios of variances are the same for i = 1, 2, and compute the error rate when the ratios λ i are 2, 3, 4 and 5. The estimated error rates are given Table 4 (see appendix) at significance level α =.5 based on 5 simulations, and the standard errors of the estimate are given the parentheses (the second row). The 95% confidence intervals for true error rate are also given the parentheses (the third row). All of the confidence intervals for the error rate for the proposed approach contain the nominal level.5, except one upper bound is lower tha.5. This indicates the true error rate could occasionally fall below the nominal level.5. The proposed method can control the family-wise error rate for different sample sizes and different ratios of variances while slightly conservative. On the other hand, some confidence intervals for TM1 do not contai.5, the lower bounds are greater tha.5. It indicates that the method can not control the error rate at a prespecified level in some cases. For TM2, all confidence intervals do not contai.5, and all the upper bounds are smaller tha.5. This indicates that the method based on Banerjee s solutios very conservative. 7 Discussion In this article, we propose an exact method for the multiple comparisons with a control without the equal variance assumption. We first derive the distribution of the test statistic, critical values and the confidence intervals of the differences between the means of the treatment groups and the control group with the known ratios of the variances. Simulation results indicate that the exact method always controls the family-wise error rate, while Dunnett s method may have an excessive error rate. Thus, it may lead to erroneous inference when the equal variance assumptios not satisfied. In practice, the ratios of the variances may not be known sometimes. To handle such a situation, we provide two approximate procedures for MCC to estimate the critical values and the confidence intervals of the differences of the means. One is using the Bonferroni approx-

11 A Multiple Comparison Procedure 175 imation and the other is using Slepian s inequality. Compared to Tamhane s approximations, our approximation with Slepian s inequality always controls the family-wise error rate for different sample sizes and variances but slightly conservative. Instead, Tamhane s approximation based on Welch s method may have inflated error rate especially when the small size is paired with large variance, therefore can not guarantee the controlling of the error rate. Thamhane s method based on Banerjee s solutions to Behrens-Fisher problems is very conservative and is not recommended. In summary, how to control the family wise error rate is a central issue in the area of the multiple comparisons. The plausibility of the equal variance condition should always be considered and verified. When the assumption of the equal variances is not satisfied, the methods with more flexible restrictions, such as the methods proposed in this article, may be considered as a more reasonable candidate for the MCC.

12 H. Li 176 Appendix In the appendix, we fist give the details of derivations of equations (13), (14) and the distribution of a i in Section 5. In addition, all the tables will be given this appendix. Derivation of Equation (13). To evaluate the critical value q, we write the following probability as antegration form. Proof. ȳ P ȳi (µ i µ ) < q, i = 1,..., k ˆσ ˆλi c + 1 = P [ȳ i ȳ (µ i µ )]/σ λi + 1 < q ˆσ, i = 1,..., k ˆσ ˆλi σ c + 1 /( λ i + 1 σ ) = Eˆσ P z i < q ˆσ, i = 1,..., k ˆσ ˆλi c + 1 /( λ i + 1 σ ) = E Eˆσ P z i < q ˆσ ˆλ i λ σ c + 1 /( λ i + 1 ), i = 1,..., k ˆσ, λ ( = G q ˆσ a1 q ˆσ ) ak p( ˆσ )f(a 1 ) f(a k )d ˆσ da 1 da k σ σ σ σ where λ = (ˆλ 1,..., ˆλ k ) and ˆλ 1,..., ˆλ k are independent given ˆσ = S. Derivation of Equation (14). First we give a lemma which will be used to derive equation (14). Lemma 5.1 (Kimball, 1951) Let X 1,..., X k be independent real valued random variables and let Ψ j (x 1,..., x k ) (1 j p) be nonnegative real valued functions each of which is nondecreasing in each of its arguments x i (1 i k). Then denoting Y j = Ψ j (x 1,..., x k ) we have p p E{ Y j } E{Y j }. j=1 j=1

13 A Multiple Comparison Procedure 177 Proof. By applying Theorem 5.1, we have [ E Eˆσ P (z i < q ˆσ ] ai, i = 1,..., k ˆσ, λ) λ σ [ k E Eˆσ P (z i < q ˆσ ai ˆσ, λ) λ σ k [ ] ˆσ E Eˆσ P (z i < d i ai ˆσ, λ) λ σ = k Φ(q ˆσ σ ai )p( ˆσ σ )f(a i )d ˆσ σ da i. ] By Slepian s inequality By Lemma 5.1 Since E λ Eˆσ [ k P {z i < q( ˆσ σ ) a i ˆσ, ˆλ } ] = E λ Eˆσ [ k Ψ i (a 1,..., a k ) ] Given ˆσ, ˆλ 1,..., ˆλ k are independent, and a 1,..., a k are also independent. In addition, Ψ 1,..., Ψ k are non-decreasing functions of a 1,..., a k. Therefore, we can apply Lemma 5.1 in the above proof. Derivation of the Distribution of a i. Proof. and a i = ˆλ i c + 1 λ i + 1 = Si 2 + S2 ( S2 i S 2 σ 2 i + σ2 = + 1 )S 2 ( σ2 i + 1 σ 2 )S 2 = S 2 σ 2 1 f i g i (S 2 i + S2 ) Si 2 + S2 ( σ 2 i ) + σ2 is approximately χ 2 f i /f i random variable based on Welch (1938). f i, g i are defined by g i = and f i can be approximated by σ 4 i n 2 i ( 1) + σ4 n 2 ( 1) σ 2 i + σ2, f i = ˆf i = ( ˆσ 2 i + ˆσ2 ) 2 ˆσ 4 i n 2 i ( 1) + ˆσ 4 n 2 ( 1) ( σ 2 i + σ2 ) 2 σ 4 i n 2 i ( 1) + σ4 n 2 ( 1),

14 H. Li 178 S 2 σ 2 χ 2 v /v, with v = 1. We have a i = (χ 2 f i /f i ) (χ 2 v /v ). Conditioning on ˆσ = S, the numerator and denominator of a i are independently distributed. Hence, a i has asymptotic F distribution with ˆf i and v degrees of freedom. Table 1: Spleen weight (in grams) of male rats Hormone Dosage Sample Mean Std. dev. (mg/kg per day) size weight weight =saline control =Oral BSA =Oral rigf-i =Oral rigf-i =Oral rigf-i =SC infusion rigf-i Table 2: Simultaneous confidence lower bounds for µ i µ (i = 1,..., 5) for α =.5 Parameter Exact Dunnett s of interest method method µ 1 µ µ 2 µ µ 3 µ µ 4 µ µ 5 µ

15 A Multiple Comparison Procedure 179 Table 3: Estimated error rate for k=2, α =.5 ( = 3, n 1 = 1, n 2 = 15) λ i Exact (.32) (.31) (.29) (.29) (.31) (.3) method (.469, (.449, (.413, (.414, (.447, (.421,.595).571).531).532).569).539) Dunnett s (.26) (.31) (.34) (.37) (.39) (.4) method (.291, (.449, (.561, (.665, (.736, (.816,.393).571).695).811).889).972) Table 4: Estimated error rate for k=2, α =.5 ( = 2, n 1 = 1, n 2 = 1) λ i Proposed method (.29) (.28) (.3) (.31) (.393,.57) (.369,.481) (.411,.529) (.46,.582) TM1 (.31) (.33) (.35) (.32) (.421,.539) (.56,.635) (.611,.749) (.484,.68) TM2 (.28) (.27) (.29) (.28) (.347,.457) (.337,.443) (.355,.469) (.355,.465 ) References [1] Bauer, P. (1997). A Note on Multiple Testing Procedures in Dose Finding. Biometrics, 53,

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17 A Multiple Comparison Procedure 181 [15] Welch, B.L. (1938). The Significance of the Diffference Between Two Means when the Population Variance are Unequal. Biometrika, 29, [16] Welch, B.L. (1947). The Generalization of Student s Problem when Several Population Variance are Involved. Biometrika, 34,